Permanent magnet brushless direct current (PMBDC) machines are well known in literature. Consider a three phase machine having three phase windings and a rotor with permanent magnets. The phase windings are balanced, in that they have the same number of turns per phase winding, and they are spatially phase shifted by 120 electrical degrees. The theory and operation of such machines are described in chapters 1 and 9 of the book, R. Krishnan, “Permanent Magnet Synchronous and Brushless DC Motor Drives”, CRC Press, 2009. The theory and operation of a three phase H-bridge inverter is common knowledge and described in the same book in chapters 2 and 10.
Consider the magnitude of a direct current (dc) input voltage to an inverter as Vs1, which can be supplied from either a battery source or a rectified alternating current (ac) source. Assuming two phase windings of a machine are series connected, the instantaneous maximum voltage that is applied across a machine winding phase through a three phase inverter is 0.5Vs1. If the current in a phase is 1, then the input power per phase is 0.5Vs1I. Since two phases conduct at any given time in a PMBDC motor drive, the maximum input power to the PMBDC machine is 2*0.5Vs1I=Vs1I.
Because the two windings are in series, they carry the same current I. From the dc link or input side, the instantaneous power supplied to the inverter and machine is the product of voltage Vs1 and current I, expressed as Vs1I. The instantaneous power is equal to the inverter input power and the machine input power, if losses in the inverter are ignored.
The inverter supplies each phase with 120 electrical-degrees-wide current in both positive and negative cycles; that is, the inverter supplies a bipolar or alternating current to the machine phase windings. Furthermore, an inverter phase leg has an ideal duty cycle of ⅔, meaning that it is active for that part of the time in the machine's ac cycle. For the remaining ⅓ of the cycle time, the inverter is in the off condition (i.e., it is not active). This is true for all three phases in the inverter.
The machine phase windings carry the alternating currents generated through the inverter, though the current of each winding is phase shifted from the others by 120 electrical degrees. The phase shift of the current in each winding is the same amount as the spatial phase shift between the winding phases, so as to produce a uniform and constant air gap torque and power.
Consider the winding resistive losses in the PMBDC machine and let the resistance per phase be Rs in units of Ohms. The total instantaneous winding power loss is equal to I2Rs per phase. Since two winding phases conduct in a PMBDC machine at any given time, the total resistive loss is 2I2Rs.